FOLLOW-UP: Learning and Making Sense of Differential Geometry in General Relativity

This post is a follow-up to the previous post [LINK] in which I tried to make sense of the mathematics of differential geometry, especially in the context of general relativity, and proposed notation that may be less confusing, more consistent, and arguably more powerful than traditional notation given the existence of a metric in general relativity. With similar motivations, this post explores how some of the ideas of electromagnetic (EM) theory may change on curved manifolds. Follow the jump to see more.

Book Review: "Atomic Accidents" by James Mahaffey

I've recently read Atomic Accidents by James Mahaffey. It's a fairly long and detailed exposition into accidents involving civilian nuclear power or military nuclear weapons in the US, UK, [former] USSR, Japan, and elsewhere. The author goes into quite granular detail with respect to the history of a certain weapon or civilian site, the timeline of the accident, and the aftermath; with each, he summarizes the lessons that were learned (or should have been learned but were not). After an introduction showing how nuclear accidents are quite similar in many respect to railroad accidents, the first few chapters go into the development of nuclear technology through WWII and the accidents along the way. The middle section of the book goes into knowledge gained about the occupational hazards of employees at power plants, the risks of transporting nuclear material by airplane, and related ideas. The last few chapters are about more recent accidents in Three Mile Island, Chernobyl, and Fukushima, ending with an assessment of the current state of nuclear power.

As a layperson with respect to the field of nuclear engineering, I think this book may be best suited for experts and other people with a serious interest in the field; for laypeople, the first few and last chapters are interesting, but reading through the middle sections became somewhat tiresome, as the technical details and jargon were a bit hard to follow, and the structure of the stories of the accidents became rather repetitive. The author does discuss issues of fear/hysteria in the general public with respect to nuclear accidents, yet the [seemingly contradictory] combination of the overall discussion of nuclear accidents in gritty detail along with some relatively cursory words of support for the safety and efficacy of nuclear power at the end means that this really is for experts who rationally understand the full historical & current contexts underlying nuclear power. Additionally, as a side note, the author briefly mentions India's nuclear program twice, noting that the Indian government broke a promise to Canada to not turn an imported reactor design toward weapons development, and that nuclear power plants there have had spotty safety records; while I am all for calling out entities that are cavalier about these issues, I found the author's word choice to be unnecessarily condescending toward India in a manner reminiscent of British imperialists justifying the subjugation of India in order to "civilize" the "savages". That aside, as noted earlier, I would recommend this book to those with a serious interest in this subject.

My Time at the 2017 APS March Meeting

This is just a quick update from my graduate studies. In the middle of March, I was able to attend the 2017 APS March Meeting in New Orleans, Louisiana; this is the first conference that I have attended in any capacity. I had a ton of fun being able to travel there and meet with old friends, collaborators whom I had not yet met in person, and other people that I didn't know but who have been working on similar things to me; it was great being able to discuss interesting potential project ideas with them. Going to many talks was also cool, giving me a good sense of the state of my field and the sort of work that other people are doing, though in many respects, I honestly feel like I learned more just from talking with people at length in more informal settings. Finally, the highlight for me was getting to present my own work and seeing the interest that many people took in it: I was presenting a semiclassical electromagnetics approach to van der Waals interactions between molecules and larger bodies in a session dominated by density functional and atomistic long-range many-body approaches to van der Waals interactions solely in [large] molecular systems, so I was truly gratified to see that members of the audience saw value and significance in what I've been doing, especially given how different my talk was from those that came before and after it in that session. Overall, I had an amazing time, and I hope to be able to attend other conferences, both similar to this as well as more specific to my research, during my remaining time in graduate school.

Reflection: My Undergraduate Experiences at MIT

Commencement is a few days away, so I don't have too much more time on campus. I've finished all four years of my undergraduate education. It has been a really wild and amazing ride, and now that things are marginally quieter, I think I could use a little reflection on those 4 years (or, at least, the highlights, learning experiences, and more recent parts that I remember). I am no poet, so a lot of this may sound repetitive, awkward, or stilted; believe me when I say this is really how I feel. Follow the jump to read more.

Green's Functions and Correlations

I had the idea of writing this post a couple of weeks ago, but I didn't feel like I had enough stuff to write here at that time. Now I do, so here goes. (Also, here's hoping that inputting LaTeX into this post works once more.)

When I took 18.03 — Differential Equations in 2010 fall, one of the topics covered was linear time-invariant systems. The general system of interest was $Lu(t) = f(t)$ where $L$ is a linear time-invariant operator. The technique of course is to find a weight function $w(t)$ where $Lw(t) = \delta(t)$, and once that is done, the solution is $u(t) = \int_{-\infty}^{\infty} f(t') w(t - t') dt'$ which is a convolution of the input $f$ with the weight $w$. The professor mentioned that it is essentially akin to inverting the operator $L$, but while I could see the general utility in this method, I never quite understood why it might be considered inversion on any deeper level.

Last semester, I took 8.07 — Electromagnetism II, and there we discussed Green's functions a little more in the context of electromagnetism & electrodynamics. In a static situation, the Green's function comes up in solving the Poisson equation $\nabla^2 \phi = -\rho$. In this case, $\nabla^2 G(\vec{x}, \vec{x}') = -\delta(\vec{x} - \vec{x}')$ is solved by the familiar potential of a unit point charge $G(\vec{x}, \vec{x}') = \frac{1}{4\pi |\vec{x} - \vec{x}'|}$. I started to see a little more clearly why this worked, because if a general charge distribution was some superposition of point charges, then a general potential distribution should be the same superposition of point charge potentials. However, it still wasn't entirely clear to me how this was "inversion" per se. Follow the jump to see what changed.

Electromagnetism Basics in 1 or 2 Dimensions

This was a post that I had been thinking of doing for a while, but I couldn't get around to it until now. A lot of introductory electricity & magnetism problems constrain charges to only move in 1 or 2 dimensions, but in reality the constraint existed within a 3-dimensional space. I thought that would cover the bases for electrodynamics in 1 or 2 dimensions, but then I saw that in cylindrical coordinates, the order-0 multipole moment outside a line charge is $\phi \propto \ln(r)$ as opposed to $\phi \propto \frac{1}{r}$. That made me realize that there is in fact a distinction among 1 or 2 or 3 dimensions. In all of the following, I will make use of the conventions and relations \[ x^{\mu} = (ct, x, y, z) \\ \partial_{\mu} = \left(\frac{1}{c} \frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right) \\ \eta_{\mu \nu} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \\ F^{\mu \nu} = \begin{bmatrix} 0 & E_x & E_y & E_z \\ -E_x & 0 & B_z & -B_y \\ -E_y & -B_z & 0 & B_x \\ -E_z & B_y & -B_x & 0 \end{bmatrix} \\ \partial_{\nu} F^{\mu \nu} = \frac{4\pi}{c} J^{\mu} \\ \epsilon_{\mu \nu \zeta \xi} \partial^{\nu} F^{\zeta \xi} = 0 \\ \mathbf{F} = q\left(\mathbf{E} + \frac{\mathbf{v}}{c} \times \mathbf{B}\right) \] in 3 dimensions, with Einstein summation and CGS implied (with more on that last point nearer to the end), with Latin indices representing only spatial components, and with Greek indices representing spacetime components. Also note that the fully antisymmetric tensor $\epsilon$ has $n$ Latin indices in $n$ spatial dimensions and $n+1$ Greek indices in $n+1$ spacetime dimensions; for example, in 2 spatial dimensions, the antisymmetric tensor over only space looks like $\epsilon_{ij}$, while over spacetime it looks like $\epsilon_{\mu \nu \xi}$, and I will frequently switch between the two as needed. Follow the jump to see what happens.

Magnetic Field Tensor and Continuum Optical Modes

There is actually a third thing in this post, but I'm not going to list that in the title. Also, the two things in the title are separate and unrelated. Follow the jump to see it all.

Schrödinger and Biot-Savart

There were two things that I would like to post here today. The first is something I have been mulling over for a while. The second is something that I thought about more recently.

Time evolution in nonrelativistic quantum mechanics occurs according to the [time-dependent] Schrödinger equation \[ H|\Psi\rangle = i\hbar \frac{\partial}{\partial t} |\Psi\rangle .\] While this at first may seem intractable, the trick is that typically the Hamiltonian is not time-dependent, so a candidate solution could be $|\Psi\rangle = \phi(t)|E\rangle$. Plugging this back in yields time evolution that occurs through the phase $\phi(t) = e^{-\frac{iEt}{\hbar}}$ applied to energy eigenstates that solve \[ H|E\rangle = E \cdot |E\rangle \] and this equation is often called the "time-independent Schrödinger equation". When I was taking 8.04 — Quantum Physics I, I agreed with my professor who called this a misnomer, in that the Schrödinger equation is supposed to only describe time evolution, so what is being called "time-independent" is more properly just an energy eigenvalue equation. That said, I was thinking that the "time-independent Schrödinger equation" is really just like a Fourier transform of the Schrödinger equation from time to frequency (related to energy by $E = \hbar\omega$), so the former could be an OK nomenclature because it is just a change of basis. However, there are two things to note: the Schrödinger equation is basis-independent, whereas the "time-independent Schrödinger equation" is expressed only in the basis of energy eigenstates, and time is not an observable quantity (i.e. Hermitian operator) but is a parameter, so the change of basis/Fourier transform argument doesn't work in quite the same way that it does for position versus momentum. Hence, I've come to the conclusion that it is better to call the "time-independent Schrödinger equation" as the energy eigenvalue equation.

Switching gears, I was thinking about how the Biot-Savart law is derived. My AP Physics C teacher told me that the Ampère law is derived from the Biot-Savart law. However, this is patently not true, because the Biot-Savart law only works for charges moving at a constant velocity, whereas the Ampère law is true for magnetic fields created by any currents or any changing electric fields. In 8.022 — Physics II, I did see a derivation of the Biot-Savart law from the Ampère law, showing that the latter is indeed more fundamental than the former, but it involved the magnetic potential and a lot more work. I wanted to see if that derivation still made sense to me, but then I realized that because magnetism essentially springs from the combination of electricity and special relativity and because the Biot-Savart law relies on the approximation of the charges moving at a constant velocity, it should be possible to derive the Biot-Savart law from the Coulomb law and special relativity. Indeed, it is possible. Consider a charge $q$ whose electric field is \[ \vec{E} = \frac{q}{r^2} \vec{e}_r \] in its rest frame. Note that the Coulomb law is exact in the rest frame of a charge. Now consider a frame moving with respect to the charge at a velocity $-\vec{v}$, so that observers in the frame see the charge move at a velocity $\vec{v}$. Considering only the component of the magnetic field perpendicular to the relative motion, noting that there is no magnetic field in the rest frame of the charge yields, and considering the low-speed limit (which is the range of validity of the Biot-Savart law) $\left|\frac{\vec{v}}{c}\right| \ll 1$ so that $\gamma \approx 1$ yields $\vec{B} \approx -\frac{\vec{v}}{c} \times \vec{E}$. Plugging in $-\vec{v}$ (the specified velocity of the new frame relative to the charge) for $\vec{v}$ (the general expression for the relative velocity) and plugging in the Coulomb expression for $\vec{E}$ yields the Biot-Savart law \[ \vec{B} = \frac{q\vec{v} \times \vec{e}_r}{cr^2}. \] One thing to be emphasized again is that the Coulomb law is exact in the rest frame of the charge, while the Biot-Savart law is always an approximation because a moving charge will have an electric field that deviates from the Coulomb expression; the fact that the Biot-Savart law is a low-speed inertial approximation is why I feel comfortable doing the derivation this way.

Nonzero Electromagnetic Fields in a Cavity

The class 8.06 — Quantum Physics III requires a final paper, written essentially like a review article of a certain area of physics that uses quantum mechanics and that is written for the level of 8.06 (and not much higher). At the same time, I have also been looking into other possible UROP projects because while I am quite happy with my photonic crystals UROP and would be pleased to continue with it, that project is the only one I have done at MIT thus far, and I would like to try at least one more thing before I graduate. My advisor suggested that I not do something already done to death like the Feynman path integrals in the 8.06 paper but instead to do something that could act as a springboard in my UROP search. One of the UROP projects I have been investigating has to do with Casimir forces, but I pretty much don't know anything about that, QED, or [more generally] QFT. Given that other students have successfully written 8.06 papers about Casimir forces, I figured this would be the perfect way to teach myself what I might need to know to be able to start on a UROP project in that area. Most helpful thus far has been my recitation leader, who is a graduate student working in the same group that I have been looking into for UROP projects; he has been able to show me some of the basic tools in Casimir physics and point me in the right direction for more information. Finally, note that there will probably be more posts about this in the near future, as I'll be using this to jot down my thoughts and make them more coherent (no pun intended) for future reference.

Anyway, I've been able to read some more papers on the subject, including Casimir's original paper on it as well as Lifshitz's paper going a little further with it. One of the things that confused me in those papers (and in my recitation leader's explanation, which was basically the same thing) was the following. The explanation ends with the notion that quantum electrodynamic fluctuations in a space with a given dielectric constant, say in a vacuum surrounded by two metal plates, will cause those metal plates to attract or repel in a manner dependent on their separation. This depends on the separation being comparable to the wavelength of the electromagnetic field (or something like that), because at much larger distances, the power of normal blackbody radiation (which ironically still requires quantum mechanics to be explained) does not depend on the separation of the two objects, nor does it really depend on their geometries, but only on their temperatures. The explanation of the Casimir effect starts with the notion of an electromagnetic field confined between two infinite perfectly conducting parallel plates, so the fields form standing waves like the wavefunctions of a quantum particle in an infinite square well. This is all fine and dandy...except that this presumes that there is an electromagnetic field. This confused me: why should one assume the existence of an electromagnetic field, and why couldn't it be possible to assume that there really is no field between the plates?

Then I remembered what the deal is with quantization of the electromagnetic field and photon states from 8.05 — Quantum Physics II. The derivation from that class still seems quite fascinating to me, so I'm going to repost it here. You don't need to know QED or QFT, but you do need to be familiar with Dirac notation and at least a little comfortable with the quantization of the simple harmonic oscillator.

Let us first get the classical picture straight. Consider an electromagnetic field inside a cavity of volume $\mathcal{V}$. Let us only consider the lowest-energy mode, which is when $k_x = k_y = 0$ so only $k_z > 0$, stemming from the appropriate application of boundary conditions. The energy density of the system can be given as \[H = \frac{1}{8\pi} \left(\vec{E}^2 + \vec{B}^2 \right)\] and the fields that solve the dynamic Maxwell equations \[\nabla \times \vec{E} = -\frac{1}{c} \frac{\partial \vec{B}}{\partial t}\] \[\nabla \times \vec{B} = \frac{1}{c} \frac{\partial \vec{E}}{\partial t}\] as well as the source-free Maxwell equations \[\nabla \cdot \vec{E} = \nabla \cdot \vec{B} = 0\] can be written as \[\vec{E} = \sqrt{\frac{8\pi}{\mathcal{V}}} \omega Q(t) \sin(kz) \vec{e}_x\] \[\vec{B} = \sqrt{\frac{8\pi}{\mathcal{V}}} P(t) \cos(kz) \vec{e}_y\] where $\vec{k} = k_z \vec{e}_z = k\vec{e}_z$ and $\omega = c|\vec{k}|$. The prefactor comes from normalization, the spatial dependence and direction come from boundary conditions, and the time dependence is somewhat arbitrary. I think this is because the spatial conditions are unaffected by time dependence if they are separable, and the Maxwell equations are linear so if a periodic function like a sinusoid or complex exponential in time satisfies Maxwell time evolution, so does any arbitrary superposition (Fourier series) thereof. That said, I'm not entirely sure about that point. Also note that $P$ and $Q$ are not entirely arbitrary, because they are restricted by the Maxwell equations. Plugging the fields into those equations yields conditions on $P$ and $Q$ given by \[\dot{Q} = P\] \[\dot{P} = -\omega^2 Q\] which looks suspiciously like simple harmonic motion. Indeed, plugging these electromagnetic field components into the Hamiltonian [density] yields \[H = \frac{1}{2} \left(P^2 + \omega^2 Q^2 \right)\] which is the equation for a simple harmonic oscillator with $m = 1$; this is because the electromagnetic field has no mass, so there is no characteristic mass term to stick into the equation. Note that these quantities have a canonical Poisson bracket $\{Q, P\} = 1$, so $Q$ can be identified as a position and $P$ can be identified as a momentum, though they are actually neither of those things but are simply mathematical conveniences to simplify expressions involving the fields; this will become useful shortly.

Quantizing this yields turns the canonical Poisson bracket relation into the canonical commutation relation $[Q, P] = i\hbar$. This also implies that $[E_a, B_b] \neq 0$, which is huge: this means that states of the photon cannot have definite values for both the electric and magnetic fields simultaneously, just as a quantum mechanical particle state cannot have both a definite position and momentum. Now the fields themselves are operators that depend on space and time as parameters, while the states are now vectors in a Hilbert space defined for a given mode $\vec{k}$, which has been chosen in this case as $\vec{k} = k\vec{e}_z$ for some allowed value of $k$. The raising and lowering operators $a$ and $a^{\dagger}$ can be defined in the usual way but with the substitutions $m \rightarrow 1$, $x \rightarrow Q$, and $p \rightarrow P$. The Hamiltonian then becomes $H = \hbar\omega \cdot \left(a^{\dagger} a + \frac{1}{2} \right)$, where again $\omega = c|\vec{k}|$ for the given mode $\vec{k}$. This means that eigenstates of the Hamiltonian are the usual $|n\rangle$, where $n$ specifies the number of photons which have mode $\vec{k}$ and therefore frequency $\omega$; this is in contrast to the single particle harmonic oscillator eigenstate $|n\rangle$ which specifies that there is only one particle and it has energy $E_n = \hbar \omega \cdot \left(n + \frac{1}{2} \right)$. This makes sense on two counts: for one, photons are bosons, so multiple photons should be able to occupy the same mode, and for another, each photon carries energy $\hbar\omega$, so adding a photon to a mode should increase the energy of the system by a unit of the energy of that mode, and indeed it does. Also note that these number eigenstates are not eigenstates of either the electric or the magnetic fields, just as normal particle harmonic oscillator eigenstates are not eigenstates of either position or momentum. (As an aside, the reason why lasers are called coherent is because they are composed of light in coherent states of a given mode satisfying $a|\alpha\rangle = \alpha \cdot |\alpha\rangle$ where $\alpha \in \mathbb{C}$. These, as opposed to energy/number eigenstates, are physically realizable.)

So what does this have to do with quantum fluctuations in a cavity? Well, if you notice, just as with the usual quantum harmonic oscillator, this Hamiltonian has a ground state energy above the minimum of the potential given by $\frac{1}{2} \hbar\omega$ for a given mode; this corresponds to having no photons in that mode. Hence, even an electrodynamic vacuum has a nonzero ground state energy. Equally important is the fact that while the mean fields $\langle 0|\vec{E}|0\rangle = \langle 0|\vec{B}|0\rangle = \vec{0}$, the field fluctuations $\langle 0|\vec{E}^2|0\rangle \neq 0$ and $\langle 0|\vec{B}^2|0 \rangle \neq 0$; thus, the electromagnetic fields fluctuate with some nonzero variance even in the absence of photons. This relieves the confusion I was having earlier about why any analysis of the Casimir effect assumes the presence of an electromagnetic field in a cavity by way of nonzero fluctuations even when no photons are present. Just to tie up the loose ends, because the Casimir effect is introduced as having the electromagnetic field in a cavity, the allowed modes are standing waves with wavevectors given by $\vec{k} = k_x \vec{e}_x + k_y \vec{e}_y + \frac{\pi n_z}{l} \vec{e}_z$ where $n_z \in \mathbb{Z}$, assuming that the cavity bounds the fields along $\vec{e}_z$ but the other directions are left unspecified. This means that each different value of $\vec{k}$ specifies a different harmonic oscillator, and each of those different harmonic oscillators is in the ground state in the absence of photons. You'll be hearing more about this in the near future, but for now, thinking through this helped me clear up my basic misunderstandings, and I hope anyone else who was having the same misunderstandings feels more comfortable with this now.

More on My Photonic Crystal UROP

In my post at the end of the summer, I talked a bit about what I actually did in that UROP. Upon rereading it, I have come to realize that it is a little jumbled and technical. I'd like to basically rephrase it in less technical terms, along with providing more context on what I did in the 2011 fall semester. Follow the jump to see more.

PCLinuxOS (Kinda) Saved My Laptop

Yesterday, I was surfing the web in my room as usual, when some exterminators came to my house and advised that I leave the room while the pesticides were applied. I unplugged my laptop from the wall outlet, forgetting that I had also removed the battery. Whoops.

Today, I was alarmed to see that I could not boot into my Linux Mint system; the OS would give a "no init found" error after the boot splash. First, I had to boot into Microsoft Windows 7; thankfully, that worked as Linux Mint was the OS I was [of course] using when I accidentally unplugged my computer. I looked up the error, and it turns out it's a common one that can be solved by a file system check ("fsck") from a live CD. All the guides I saw recommended using a live CD of the same OS whose hard drive partition is affected, but I had left my live CDs and USB sticks in my dormitory room. Whoops again. What I figured would just be a minor inconvenience turned into a semi-major problem.

First I tried making a Linux Mint live CD from within Microsoft Windows 7. The first attempt failed because I didn't use the right tool to burn the ISO file to the CD. The second attempt failed too, but that is because the live CD would hang during the [very long] boot process after the boot splash screen was done.

After that, I figured it may just be an issue with Linux Mint, so I rummaged through some CDs and found an old Fedora 11 "Leonidas" GNOME live CD. That would also hang right after the boot splash, so at that point I became quite worried that my computer may never be able to work with Linux ever again.

Finally, I found an old PCLinuxOS 2009.2 KDE live CD, and I gave that a shot. To my relief, it worked, although it would not connect to the Internet. I entered "su" into the terminal followed by "fsck -y /dev/sda5" (because my Linux Mint partition is "sda5"), and that seemed to work OK. Rebooting into my installed Linux Mint system worked! I was incredibly happy to see my main OS back in working order and to see that I would not have to resign myself to using Microsoft Windows 7. There are only two lingering issues, and both of them are quite minor. The first is that some odd error message involving "PXE" and "PCI" is displayed for a few seconds just before the GRUB boot menu is displayed, but that goes away on its own anyway. The second, which may be related to the first, is that Linux Mint is apparently no longer able to recognize my laptop's ethernet card, so I must now depend on wireless connections to access the Internet, and I will have to train myself to overcome my aversion to wireless connections stemming from numerous failed attempts by my family to send wireless Internet signals to the very room where I am typing this. Anyway, regardless of the fact that I don't particularly care for the newer versions of PCLinuxOS, I salute [the older version of] PCLinuxOS for doing its job when I needed it most!

(UPDATE: I was a little concerned that this accident may have fried my laptop's ethernet card, so as per some advice posted somewhere online, I unplugged my laptop from everything (power, ethernet, even the USB mouse), and didn't touch it again for the next 16 hours (i.e. from yesterday afternoon until now). For one, that weird error message has disappeared. For another, the ethernet card is now recognized in the "System Profiler and Benchmark" program as well as in the output of the terminal command "lshw -C network", although my laptop refuses to connect to the ethernet. Hooray!)